Estimating label probability bounds:  We randomly sample \(n\)  Gaussian noise from \(\epsilon \)  and add them to the input \(\mathbf {x}\) .
We use \(\mathbf {x}^1, \mathbf {x}^2, \cdots , \mathbf {x}^n\)  to denote the \(n\)  noisy inputs for convenience. Given these noisy inputs, we use the base multi-label classifier \(f\)  to predict \(k^{\prime }\)  labels for each of them.
Moreover, we define the label frequency \(n_i\)  of label \(i\)  as the number of noisy inputs whose predicted \(k^{\prime }\)  labels include \(i\) . Formally, we have \(n_i = \sum _{t=1}^n \mathbb {I}(i \in f_{k^{\prime }}(\mathbf {x}^t)), i \in \lbrace 1,2,\cdots ,c\rbrace \) , where \(\mathbb {I}\)  is an indicator function.
Based on the definition of label probability \(p_i\) , we know that \(n_i\)  follows a binomial distribution with parameters \(n\)  and \(p_i\) , where \(n\)  is the number of noisy inputs and \(p_i\)  is the label probability of label \(i\) . Our goal is to estimate a lower or upper bound of \(p_i\)  based on \(n_i\)  and \(n\) , which is a binomial proportion confidence interval estimation problem. Therefore, we can use the Clopper-Pearson [1]} method from the statistics community to estimate these label probability bounds. Formally, we have the following label probability bounds:

\(& \underline{p_i} = \text{Beta}(\frac{\alpha }{c}; n_i, n - n_i + 1), i \in L(\mathbf {x}),\\& \overline{p}_j = \text{Beta}(1-\frac{\alpha }{c}; n_j, n - n_j + 1), \forall j \in \lbrace 1,2,\cdots ,c\rbrace \setminus L(\mathbf {x}),\) 
